y=(xy)x Write an expression for dy/dx

Question

y=(xy)^x Write an expression for dy/dx

Answer 1

Take natural log of both sides.

 

ln(y) = x[ln(x) + ln(y)]

 

Differentiate.

 

y'/y = [ln(x) + ln(y)] + x(1/x + y'/y)

 

The rest is just algebra.  Can you solve for y'(x,y) and finish from here?

Answer 2

Use logarithmic differentiation:

 

y = (xy)^x

 

lny = ln(xy)^x

 

lny = xln(xy) = x(lnx + lny)

 

y'/y = lnx + lny + x(1/x + y'/y)

 

y'/y - xy'/y = lnx + lny + 1

 

y'[(1-x)/y] = ln(xy) + 1

 

           y' = y[ln(xy) + 1] / (1-x)

 

           y' = (xy)^x(ln(xy) + 1) / (1-x)